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Talk:Bird's array notation
Plugging array-like notations to functions I have the plan how to adopt Bird's array notation (or probably any recursive notation) to any other function. Just change some main rules: Rule M1: (no separators) \(\{a\} = \Sigma(a)\) Rule M2: (first entry is 1) \(\{1,\#\} = 1\) Rule M6: (string of 1's from 2rd entry to nth) \(\{a,1,1,\cdots,1,1,c \#\} = \{a,a,a,\cdots,a,\{a-1,1,1,\cdots,1,1,c \#\},c-1 \#\}\) Rule M7: (otherwise) \(\{a,b \#\} = \{\{a-1,b \#\},b-1 \#\}\) All other rules remain unchanged. The limit ordinal of this notation is \(\omega_1^{CK}+\theta(\varepsilon_{\Omega+1})\), in other words, Bachmann-Howard ordinal-typed recursion around busy beaver function. There will be also more powerful variant of H(n) function (that grows on par with \(f_{\omega_1^{CK}+\theta(\varepsilon_{\Omega+1})}(n)\)). In this notation, H(1) = \(\Sigma(3) = 6\) and H(2) will be certainly larger than anything we can write down in the observable universe, using recursive definitions. In general, we can plug any so-far-defined recursive notation around any so-far-defined function, if we replace the terminating rule of the notation to computing function and probably change some other rules. If the function has ordinal level \(\alpha\) and the notation \(\beta\), then super-notation will has the ordinal level \(\alpha+\beta\). Ikosarakt1 (talk ^ ) 12:28, May 25, 2013 (UTC) That's cool,but isn't it weak for uncomputable functions?I mean everybody tries to increase ucomputable functions by adding computable notations inside.But that way you'll never boost an uncomputable function to the next uncomputable function after it.Boboris02 (talk) 13:24, November 3, 2016 (UTC) New Expansion Pack - Ready for downloading! The new Expansion Pack for Battle of Array Nations, dubbed "Newer Subscripting Adventure Network", is now available for download! This Expansion Pack adds new quests, with a new super-strong final boss, called \(\theta(\theta_1(\Omega))\). So yeah, download it on hereChanged URL as it got usurped. ☁ I want more ⛅ 02:58, June 21, 2013 (UTC), now! Have fun!dammit DrCeasium was faster -- ☁ I want more ⛅ 10:31, May 28, 2013 (UTC) Define CB(t) as function time->Ord defining upper bound of current Chris Bird's work. I state CB(t) eventually outgrows every recursive function. Proof of this would make Church-Turing thesis false. So waiting for results. LittlePeng9 (talk) 17:49, May 28, 2013 (UTC) :The problem is that Bird can eventually exhaust all symbols invented by mankind and all possible well-defined recursive combinations of them. I wonder whether happen earlier: this point or his numbers will be bigger than \(\Sigma(1000000000)\). Ikosarakt1 (talk ^ ) 16:02, June 10, 2013 (UTC) ::If Bird uses all symbols mankind invented, he will invent new ones! And this will happen waaay before we reach even like S(10000). LittlePeng9 (talk) 20:51, June 10, 2013 (UTC) ::Nonetheless, as the number of symbols becomes too large, say, million of symbols, it is hard even to distinguish some of them, let alone parse how to solve arrays using them. For example, normal backslash and backslash rotated to the right by 1 degree. Currently, he use backslash as a main symbol allowing so big level of recursion, but I feel that soon he will use a new symbol since all combinations on \ will be used. Ikosarakt1 (talk ^ ) 12:06, June 11, 2013 (UTC) ::So how about using some arbitrary function c(n) to denote characters? Just like Bird did to replace negation sign, diamond and sun with \(\backslash_n\)? But, as I guess, for any such numbering he will further diagonalize through it to reach further, and now I ran out of ideas. LittlePeng9 (talk) 15:33, June 11, 2013 (UTC) ::Yes, he can diagonalize it further and further, but at the some point (when all subscripts, superscripts, etc. will be used) in order to avoid ambiguousness he must invent the new symbol that can't be expressed in terms of previous. It already happened with nested array notation, after which he made backslash separator. I believe that c(n) can denote n-th principially new separator. c(1) is comma, c(2) is backslash and c(3) probably will be shown in Bird's next part. Ikosarakt1 (talk ^ ) 16:03, June 11, 2013 (UTC) :Eagerly awaiting the release of the Ordinals server mod. FB100Z • talk • 20:20, June 10, 2013 (UTC) Chris Bird has extended his notation again, and this time he claims to reach \(\theta(\Omega_{\omega}, 0)\)! It should appear on his web page shortly. Deedlit11 (talk) 05:42, June 13, 2013 (UTC) :What kind of pure awesomeness is this? :Oh, FB100Z, if you can read this, please also give Bird a sandwich! They both live in Britain, so it wouldn't be too hard! -- ☁ I want more ⛅ 06:56, June 13, 2013 (UTC) :The Even Newer Expansion Pack seems promising. I'll wait for it! -- ☁ I want more ⛅ 07:13, June 13, 2013 (UTC) :2 days and still waiting... -- ☁ I want more ⛅ 06:32, June 15, 2013 (UTC) :It's been a week now. What happened to Bird's new array notation? -- ☁ I want more ⛅ 04:50, June 20, 2013 (UTC) ::Sorry, we just have to wait for Munafo to update the web page. Have patience! Deedlit11 (talk) 05:02, June 20, 2013 (UTC) Nested separators Where I was wrong? A = [1 [2 2] 3] B = [1 [2 2] 2] Step 1: A_1 = [1 [2 2] 3] B_1 = [1 [2 2] 2] Step 2: A_2 = [1 [2 2] 3] B_2 = [1 [2 2] 2] Step 3: Lev(A_2) = 2 Lev(B_2) = 2 Lev(A_2)=Lev(B_2)>0 Step 4: A* = [2 2] B* = [2 2] A*=B*=H Step 5: Num(H,A_2)=1 Num(H,B_2)=1 Num(H,A_2)=Num(H,B_2) Step 6: A_2 = 1 B_2 = 1 Step 3: Lev(A_2)=0 Lev(B_2)=0 Lev(A_2)=Lev(B_2)=0 Step 7: 1=1 Step 8: A_1 = 1 B_1 = 1 Step 2: A_2 = 1 B_2 = 1 Step 3: Lev(A_2) = 1 Lev(B_2) = 1 Step 4: No separators? Ikosarakt1 (talk ^ ) 14:17, June 10, 2013 (UTC) I think the problem came from the vaguity of Step 6: what means "remove all entries up to and including H separator"? There are two ways to interpret it, and given something like [2 2], after Step 6 it can be either 2 or 1,1,1,2. Ikosarakt1 (talk ^ ) 15:58, June 10, 2013 (UTC) A Review of the New Release of BAN The new release, released today (AFAIK), was both astonishing and disappointing. It was not a further extension of the previous release, which I wanted, but rather an overhaul of the old release, with the core functions re-designed and the support for array subscripts being dropped. Yet it enables us to reach previously inaccessible dungeons, such as the Space of Clueless Galaxies. Also, it looks a lot similar to the type-''n'' brackets of HAN. So, what's your thoughts? -- ☁ I want more ⛅ 03:40, June 21, 2013 (UTC) I've some problems with opening .pdf files, so I'm waiting when HTML version of this stuff will appear. Ikosarakt1 (talk ^ ) 09:59, June 21, 2013 (UTC) Anyways, I can open it (with glitches, but can) and I concluded that it is essentially Nested-Hyper Nested Array Notation that allows to mix n-hyperseparators on different hypernesting levels. I wonder how far Bird will able to go with Hierarchical Nested-Subscript Array Notation. Ikosarakt1 (talk ^ ) 12:43, June 21, 2013 (UTC) It looks like it works. By that I mean I scrolled through it and concluded it would probably be quite a lot easier to just trust Bird on this. DrCeasium (talk) 12:47, June 22, 2013 (UTC) My thinking is, redesigning your notation to make it stronger without adding extra complexities is a GOOD thing, better than if he needed to extend it to reach the same level. Note that he can now add his previous extensions and reach an even larger ordinal. Deedlit11 (talk) 09:49, June 24, 2013 (UTC) Also, Bird's array subscripts are not similar to the type-k brackets in HAN. In HAN, []_{k+1} is just n nestings of []_k, whereas in Bird's notation, each new subscript is a generalization of the one before it. I believe Bird's subscripts are much stronger. The only similarity really is that both claim that n-subscripts reach \(\theta (\Omega_{n+c})\) in the FGH. Deedlit11 (talk) 10:10, June 24, 2013 (UTC) Loader's ordinal Any guesses about the number of part in which Bird will reach Loader's ordinal? Ikosarakt1 (talk ^ ) 10:03, June 21, 2013 (UTC) :Do we even know what that ordinal is yet? DrCeasium (talk) 12:34, June 22, 2013 (UTC) :No, and this is whole point! It is beyond notations we know, but we might hope someone (e.g. Bird) will develop suffiscently strong one. LittlePeng9 (talk) 12:41, June 22, 2013 (UTC) :If my most recent addition to HAN (new one put out today) doesn't reach it, I will be quite suprised. (its not quite finished yet, still working on the final definition of some of the rules). Not that there would really be any way of knowing, seeing they're both now beyond any ordinal notation. DrCeasium (talk) 12:50, June 22, 2013 (UTC) :In my opinion, your newest addition looks a bit like simple diagonalization. I mean, | diagonalizes through subscript arrays. ┆ diagonalizes through |-spaces. ⁅_k diagonalizes through ⁅_k-1. I might be wrong, but I think calculus of constructions (the thing Loader.c uses) can define such diagonalizations, and even beyond. But we won't know that until someone will give any upper bound on Loader's ordinal, or suffiscently large lower bound (to show which notation is stronger). LittlePeng9 (talk) 13:22, June 22, 2013 (UTC) :Yes it is reasonably simple diagonalization, but look how far that got Wythagoras (all the way up to the Bachmann-Howard and a long way beyond), and because the notation already has overtaken the level of Omega_n in the ordinal collapsing functions, that makes this diagonalisation a lot more powerful. Also, it is still under development, and so don't consider anything in this extension final yet. DrCeasium (talk) 13:28, June 22, 2013 (UTC) ::The fact that it is simple diagonalization is precisely the reason not to believe that your notation has overtaken the level of Omega_n in the ordinal collapsing functions. Deedlit11 (talk) 10:01, June 24, 2013 (UTC) :I think main strength of your notation might come from basic concept, but this may not really grow that fast later. Consider Bird's work for analogy (this part). Simple nested [] arrays reach e_0. By adding [[]] which diagonalize through nested []'s and nesting then we can then reach e_1, which is quite a step. With [] we can extend this result to e_2 etc. It can be analoguous to your [_3 etc. By further diagonalizing (\A in separators can be analogue of [_A) we can go all the way to phi(w,0). But compared to other notations it isn't that big ordinal, and same may happen to your notation, if you use that simple diagonalizations. [[User:LittlePeng9|LittlePeng9] (talk) 15:51, June 22, 2013 (UTC) :I suppose its not the best approach, but still, Bird only uses single numbers in the subscript, but I think the way mine works is quite significantly more powerful, due to the sheer number of diagonalisations. I will probably look for a better method soon, but for now it will do, as it still keeps me a very long way ahead of anybody else except for Loader. The trouble is making an array notation more powerful than CoC, which seems to be impossible to describe without referencing hundreds of other similarly complex calculuses. DrCeasium (talk) 17:35, June 22, 2013 (UTC) ::I don't understand how you can think that your subscripts are significantly more powerful than Bird's. As LittlePeng9 pointed out, if Bird defined his subscripts the way you did, []_{n+1} would reach epsilon_n in the FGH. The way Bird defines it, it goes much further, as / 2 already works as your []_X does, and his notation goes much further than that. Deedlit11 (talk) 10:01, June 24, 2013 (UTC) ::And it is debatable that HAN is really more powerful than BEAN. After my latest revision on it, BEAN goes well beyond Bachmann-Howard ordinal. Ikosarakt1 (talk ^ ) 10:12, June 24, 2013 (UTC) :The main trouble comes from understanding the CoC itself. Wikipedia, as well as other sources I've ever found on this subject, can't give much sense of it. It is extremely hard to imagine how Deedlit and LittlePeng were able to program all this and diagonalize in the formal language. Am I so stupid? Ikosarakt1 (talk ^ ) 18:52, June 22, 2013 (UTC) :Umm, Ralph Loader on his site has original version of his program, which is easily readable. Only problem I had is that compiler I use doesn't quite work with one type of recursion, so I had to modify it slightly, but that's all. If you understand lambda calculus, you'll understand Loader's program. LittlePeng9 (talk) 19:13, June 22, 2013 (UTC) Uncomputable array notation How about making array notation at level \(\omega_1^\text{CK}\) or even beyond? Ikosarakt1 (talk ^ ) 09:18, June 24, 2013 (UTC) I found pretty simple one, with only one rule: \(\{a,...\}=\Sigma(a)\) where ... is any number of variables. LittlePeng9 (talk) 21:06, June 27, 2013 (UTC) :Can we make it without references to Busy Beavers and Turing machines? Ikosarakt1 (talk ^ ) 18:02, November 11, 2013 (UTC) ::Well, if by "no Turing machines" you allow other models of computation then obviously yes. Otherwise, we can do it by allowing rule of sort "if B has a value and something something, then A reduces to B". That way we can arithmetically simulate TM and know if machine halts, thus evaluating BB(n). I'm almost sure this will work. LittlePeng9 (talk) 18:24, November 11, 2013 (UTC) :::Using deedlit11's ordinal BEAF, we can define a ω1CK structure. 15:33, August 2, 2015 (UTC) Hierarchical Nested Subscript Array Notation Any guesses about its strength? I believe it can get to the natural limit of theta function (first \(\alpha\) such that \(\theta(\Omega_\alpha) = \theta(\alpha)\)) already at [1 \\ 3 2]. There is a good chance that the limit of this notation will be larger than anything expressable using strong systems like HAN, BEAN and Dollar's function. I guess that, in order to continue further, Bird will use Taranovsky's notation and I hope he will explain it more comprehensibly. Ikosarakt1 (talk ^ ) 14:59, June 27, 2013 (UTC) :Do you mean \(\theta(\Omega_{\Omega})\) or \(\theta(\Omega_{\Omega_{\Omega...}})\)? DrCeasium (talk) 16:10, June 27, 2013 (UTC) HAN's at that ordinal with only 2. All the further (and admittedly slightly naïve) extensions are at over the level of subscripted Omegas in the theta function and so ought to be completely inexpressible in the theta function. I'm still looking for a good way to extend my notation in a better way, and I'm considering the possibility of diagonalising over the process of defining arrays itself (currently not really working). DrCeasium (talk) 16:02, June 27, 2013 (UTC) :Note that, if your notation up to X has ordinal \(\alpha\), and X has ordinal approximately \(\theta(\Omega_k, 0)\), then [_X X] will go up to ordinal \(\theta(\Omega_{\alpha}, 0)\), [_{[_X X]} X] will go up to ordinal \(\theta(\Omega_{\theta(\Omega_{\alpha}, 0)}, 0)\), and so on. The limit ordinal of that sequence is \(\theta(\Omega_{\Omega}, 0)\), so 2 would have ordinal \(\theta(\Omega_{\Omega}, 0)\), not (\theta(\Omega_{\Omega_{\Omega...}}, 0)\). Deedlit11 (talk) 03:32, June 28, 2013 (UTC) :I think the rules aren't very clear for this part, but the way I wan't it to work is to reach theta(alpha). I think the rule may need a small addition before I reach there though. Maybe double subscripts (X ~ Omega_Omega) or something. DrCeasium (talk) 14:45, June 28, 2013 (UTC) I think that Ultimate Array Notation II reach \(\theta(\alpha)\) or \(\theta(\Omega_{\Omega_{\Omega...}})\) Wythagoras (talk) 18:52, June 27, 2013 (UTC) Wait, the notation that can be achieved from combining Bird's last two ones can reach the limit of theta function much earlier: at /[1 ~ 2 2]. It follows from the principle that exhausting one type of operator launches the next type - so /[1 ~ 2 2] is limit for /[2 2], /[1 /[2 2] 2], /[1 /[2 2] 2] 2], etc... Ikosarakt1 (talk ^ ) 18:04, June 28, 2013 (UTC) :But, ~ 2 is equivalent to /_2 2 and so is less than /[2 2]. So /[1 ~ 2 2] is less than /[1 /[2 2] 2]. Deedlit11 (talk) 18:56, June 28, 2013 (UTC) :: /2 is a 1-hyperseparator, but ~ or /_2 is a 2-hyperseparator. Therefore, ~ has higher level than /2. -- ☁ I want more ⛅ 01:01, June 29, 2013 (UTC) ::Also, the separator \1¬2 is disallowed in the Nested Subscript Array Notation, because it can be rewritten as \\ 2, and since ¬ is a 2-hyperseprarator and \\ is a (1,2)-hyperseparator, it is not possible to place both ¬ and \\ on the same 'nested level'. ::However, on our new notation, this restriction is eliminated, and the separator /1~2 is valid. -- ☁ I want more ⛅ 01:16, June 29, 2013 (UTC) ::: A 1-hyperseparator is not necessarily less than a 2-hyperseparator. Assuming we are using Bird's old Beyond Nested Arrays IV, /2 is defined by ::: {a, b /[2 2] 2} = {a < 1 /2 2> b} = {a ... b> b} :::so /2 will be greater than all /_b hyperseparators. Deedlit11 (talk) 01:21, June 29, 2013 (UTC) ::::The separator //[2 2] and /1~2 can be rewritten as /[2 2 // 2] and ~ 2 // 2, respectively. Since /2 is a 1-hyperseparator and ~ is a 2-hyperseparator, and we have to use all 1-hyperseparators available before we can use 2-hyperseparators, so ~ 2 // 2 has higher level than /[2 2 // 2]. -- ☁ I want more ⛅ 01:24, June 29, 2013 (UTC) :::::/2 is definitely at a higher level than ~, since /2 resolves to /_b for arbitrary b and ~ is /_2. I suppose the order could reverse with /[2 2 // 2] and ~ 2 // 2, but it seems odd that that would happen. Could you point me to where it says in Beyond_Nested_Arrays_Old.pdf that we have to use all 1-hyperseparators available before we can use 2-hyperseparators? Deedlit11 (talk) 01:01, June 30, 2013 (UTC) ::::I think, using Hierarchical NSAN, current /[2 2 // 2] can be rewritten as [1 /[2 2] 2 // 2], and /[2 2 // 2] can represent something larger: as /2 definitely more powerful separator than ~. Ikosarakt1 (talk ^ ) 09:29, June 29, 2013 (UTC) :::::I found [12/1,222] has level that growth rate U() function and [1/22/1,222] has level that e(growth rate U() function+1). AarexTiao 14:34, January 1, 2014 (UTC) Now that the official extension (the Nested Hierarchical Hyper-Nested Array Notation) has been released, it turns out that /[1 /[2 2] 2] has level θ(Ωω, θ1(Ωω)), and /[1 /2 2 2] has level θ(Ωω, Ω2). Looks like I am the winner was correct after all! For some reason, people tend to overestimate the strength of extensions like this. They thought it would go up to collapsings of Mahlo cardinals, and it actually only goes up to θ(ΩΩ). -- ☁ I want more ⛅ 15:08, April 4, 2014 (UTC) : Everything is still ahead of Bird! :D LittlePeng9 (talk) 15:15, April 4, 2014 (UTC) I am thinking if Bird is ever going to release Beyond Bird's Nested Arrays VI. The θ(ΩΩ)-level separator is almost certainly [2 /1 / 2 2 2]. I can even go further and find that [2 /1 / 3 2 2] has level θ(ΩΩ2) and [2 /1 /2 2 2 2] has level θ(ΩΩ2). Is /1,2 really inside two pairs of square brackets or is it inside an infinite number of square brackets? So I went ahead and invented a new rule: If Xm is an m-hyperseparator and the whole thing is a k-hyperseparator and m is greater than k (they can be arrays too), [Xm 2]k collapses to Xk. I call this the collapsing rule. So we can have things like [1 [1 [6 /8 3 2] 2] 2] = /8 34. By reversing this rule, I find [2 /1,2 2 2] = [1 [1 [1 [1 [ ... [2 /1,2 2 ... ]5 2]4 2]3 2]2 2] 2]. This indicates that /1,2 is inside an infinite number of square brackets and the two square brackets is only indirectly. Following this logic, /1,3 must be inside ω2 pairs of square brackets, /1,1,2 inside ω^2 of them, and so on. This also shows that [1 /3 2 /1,2 2 2] is a valid separator because the /3 is inside ω pairs of brackets (since it's on the same nested level as /1,2, they must both be inside the same number of brackets!). Also, my large number videos are going to show that! Dshamlin (talk) 16:08, December 17, 2018 (UTC) page is unfinished the bird's array notation page isn't currently finished because of Nested Subscript Arrays, Hierarchical Hyper-Nested Arrays, and Nested Hierarchical Hyper-Nested Arrays 01:45, April 4, 2014 (UTC) :i don't know much about BAN, so can Ikosarakt/Peng/Wyth do this? FB100Z • talk • 21:47, March 30, 2014 (UTC) ::The formal ruleset for it is much too complicated to write down. So I haven't explored it in much details so far. Ikosarakt1 (talk ^ ) 14:18, March 31, 2014 (UTC) The real strength of different notations I see a lot of notations in blogs of our members that are easy, without any problems, it is not straining, with a minimum of rules, reach cardinal Mahlo, and then not only continue due to lack of even stronger ordinal notation. They are wrong and their notations are not so strong? because Bird professional mathematician, we see how much needs to be done to reach at least until the Takeuti-Feferman-Buchholz ordinal. Who is right???? Konkhra (talk) 22:13, April 4, 2014 (UTC) :When evaluating an angle bracket in Bird's array notation, all work is done in the angle bracket and you don't need any information outside the bracket apart from the two numbers on the left and right of it. On the other hand, notations like Dollar function and R function, that operate similar to the Buchholz hydra, requires looking at a parent expression to evaluate a certain expression. I think that's what makes the difference. -- ☁ I want more ⛅ 23:51, April 4, 2014 (UTC) :I think our notations indeed aren't really strong. In the list of comparisons, (infinitely) many ordinals have been omitted, only "major milestones" have a place. Nobody cares about something like \psi(\Omega_{\Omega_2+\psi(\Omega)}+\omega^{\Omega+1}) , but if it can't be expressed in the notation which claimed to go to \psi(\psi_I(0)) , this is certainly wrong claim. Bird's notation was managed to cover all possible cases up to \psi(\Omega_\Omega) , it creates full system. Ikosarakt1 (talk ^ ) 05:38, April 5, 2014 (UTC) :: Do I understand correctly, that if we consider all the ordinals, as it does Bird, BEAF not be as strong as we think now? Konkhra (talk) 05:44, April 5, 2014 (UTC) ::BEAF hadn't any strength initially, Bowers left us with some milestone definitions like {a,b / 2} = a & a & a ... a & a & a (b a's), but he said exactly nothing about how to solve 2*3^{3,3,3,3} & 3. So "true" definition doesn't exist, we can define it as strong as we want. Ikosarakt1 (talk ^ ) 05:55, April 5, 2014 (UTC) :::Well, the problem with BEAF is that it is a totally vague system, no formal definition at all. So there are a lot of different ways one could go about giving a formal definition to BEAF, and it's hard to say which one is the "right" one. (So I disagree with Sbiis Saibian that formalizing BEAF is the "holy grail" of mathematics, it's too inherently subjective.) Hyp Cos made the decision that the @ operator took a function at the level of \(G_\alpha\) to the level of \(F_\alpha\), i.e. the slow-growing hierarchy to the fast-growing hierarchy. That's an interesting formulation, but there's no evidence that Bowers even knew about these hierarchies, or about ordinals at all. Certainly passing from the slow-growing to the fast-growing hierarchy is a very strong notion and would lead to very large numbers, so Hyp Cos's version of BEAF is very strong. Despite his efforts at unravelling the "catching function", though, there is no proof of his results, and in professional mathematics these results are still considered unsolved. Of course, the growth of the slow-growing hierarchy has proven to be very sensitive to the definition of the fundamental sequences, so there may be no canonical answer. :::Note that, while Hyp Cos's version of BEAF is very strong, it is different from other notations in that it skips over ordinals, and isn't defined by recursion or iteration over previous functions. So it may be considered "cheating", although if you are just interested in large numbers you won't care about that. :::Concerning other notations such as the Dollar function and the R function, as they are based on the Buchholz hydra we can see that they pass \(\psi(\Omega_\omega)\), and they certainly appear to pass \(\psi(\psi_I(0))\). But as for the ease with which they pass Mahlo and weakly compact cardinals and beyond, I am still not convinced. Deedlit11 (talk) 06:07, April 5, 2014 (UTC) ::::If the members of our community relatively easy to create such a strong notations until compact cardinals and beyond, you have not changed your opinion that the notation that can reach BB(1000) can not be created? Konkhra (talk) 07:44, April 5, 2014 (UTC) ::::In order to define large number, we have to define ordinals namely from previous ones, giving an explicit definition of fundamental sequences. If a number has no completely explicit definition without any ambiguities, then it is not a number. We can't admit, for example, that f_{\omega_1}(3) is a number: there is no way to define \omega_1 -th function consistently. Ikosarakt1 (talk ^ ) 13:45, May 16, 2014 (UTC) Here are some comparisons between BEAF and FGH I could find in the history of googology: There's probably more of these. -- ☁ I want more ⛅ 02:54, April 7, 2014 (UTC) Eh? What is the separator [122] in terms of 1,2, 1,1,2, 1,1,1,2, etc.? Is it 1,1,1,...,1,2 with some certain number of 1's? Billicusp (talk) 20:09, March 1, 2016 (UTC) :Since the \begin{eqnarray*} \{b,p[122]2\} \\ = \{b‹022›p[122]1\} \\ = \{b‹022›p\} \\ = \{b‹p‹1›p21›p\} \\ = \{b‹p‹1›p›p\} \\ = \{b‹\underbrace{p,p,\cdots,p}_p›p\} \end{eqnarray*} :, and \begin{eqnarray*} \{b,p\underbrace{1,1,\cdots,1}_p,22\} \\ = \{b‹\underbrace{0,1,\cdots,1}_p,2›p\underbrace{1,1,\cdots,1}_p,21\} \\ = \{b‹\underbrace{p,p,\cdots,p}_p,1›p\} \\ = \{b‹\underbrace{p,p,\cdots,p}_p›p\} \end{eqnarray*} :, so \([122]\) and \(\underbrace{1,1,\cdots,1}_p,2\) is the same separator (where p is the second item of the array). Khankao1 (talk) 06:56, March 3, 2016 (UTC) Thank you very much! Billicusp (talk) 13:13, March 3, 2016 (UTC)